In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

中文翻译：

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两个分量太简单了：具有三个分量的振荡 Fisher-KPP 系统示例

在 Cantrell 最近的一篇论文中等。[9]，分析了具有 Lotka-Volterra 类型竞争的双分量 KPP 系统，并在很大程度上解决了它们的长期行为。特别是，作者确定，任何恒定的正稳态，如果是独特的，必然具有全球吸引力。在本文中，我们给出了一个明确且生物学上非常自然的振荡三分量系统示例。使用基本技术或预先建立的定理，我们表明它具有独特的恒定正稳态，具有二维不稳定流形，稳定的极限环，接近稳态的捕食者 - 猎物结构，周期性波列和点对点周期性快速行波。在数值上，我们还展示了脉动前沿和传播阶地的存在。